An instability of the Godunov scheme

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Abstract

We construct a solution to a 2 × 2 strictly hyperbolic system of conservation laws, showing that the Godunov scheme [13] can produce an arbitrarily large amount of oscillations. This happens when the speed of a shock is close to rational, inducing a resonance with the grid. Differently from the Glimm scheme or the vanishing-viscosity method, for systems of conservation laws our counterexample indicates that no a priori BV bounds or L 1-stability estimates can in general be valid for finite difference schemes.

Original languageEnglish (US)
Pages (from-to)1604-1638
Number of pages35
JournalCommunications on Pure and Applied Mathematics
Volume59
Issue number11
DOIs
StatePublished - Nov 1 2006

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

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